Vayetzei- Sorting, Sets, Subsets, and Unions related activity ideas

"[Lavan] said, 'What shall I give you?' And Yaakov said, 'Do not give me anything; if you will do this thing for me, I will resume pasturing and guarding your flocks: Let me pass through your whole flock today. Remove from there every speckled or dappled lamb, every brownish lamb among the sheep and the dappled or speckled among the goats- that will be my wage. Let my integrity testify for me in the future when it will come regarding my wage before you; any that is not speckled or dappled among the goats, or brownish among the sheep, is stolen in my possession.'
And Lavan said, 'Yes! If only it will be as you say.'
So he removed on that day the ringed and dappled he-goats and all the speckled and dappled goats- every one that had white on it, as well as all the brownish ones among the sheep- and he put them in the charge of his sons. And he put a distance of three days between himself and Yaakov; and Yaakov tended Lavan's flock that remained." ~Bereishit 30;27-36

In this week's parsha, we learn of how Lavan paid Yaakov for his years of work by separating his flocks into two groups. Lavan kept for himself all of the sheep and goats that were completely white or completely brown, and he gave to Yaakov all of the sheep and goats that had any kind of mixture of white and brown, regardless of the patterning- speckled, dappled, ringed, etc.

Activity Connections:
The concept of separating is a basic one that we begin working on with students at the preschool level. It is often thought of just as a primary skill, but the understanding of sorting, separating, and comparing groups is one that is necessary through higher levels of mathematics, as students work with logical thinking, and sets and subsets of information.

Activity ideas in increasing conceptual difficulty:
*The most basic level of sorting items is creating two or more simple categories and separating the items into the different categories. If I have a box of buttons, I might choose to sort them by color (green, blue, red, etc.), by size (large, medium, small), by shape (round, square, other), by number of holes (1, 2, 4, etc), or any other distinguishing feature that I notice. To make it even simpler, you can create just two categories- green & not green, large & small, round & not round, 1 or 2 holes & 3 or more holes. Sometimes it's easiest to just create two categories, and sometimes it's helpful to have the additional categories. This decision will be based on the items that you're using, if there's a purpose for the sorting beyond just sorting (do you need specific items for different projects?), and the way in which the students see the groupings of items that they're trying to sort. Even at an early age, you can see aspects of how each child processes information based on sorting decisions that they make when categorizing items in a group.

*The next step in sorting complexity is learning how to deal with items that might fall in two categories at the same time. The ultimate model for this situation is a Venn diagram. Students can practice sorting out actual items on an oversized mat that has a Venn diagram drawn onto it, or you can use string or hula-hoops to create overlapping circles for them the place objects into. This works for children as young as older preschool students. As students get up into early elementary classes, they should be able to record their work on a Venn diagram map, ultimately not needing to actually physically move items around. 
--When using Venn diagrams for sorting, you begin crossing categorizations. It could be straightforward- maybe some of your buttons are solid colors and others have multiple colors. Blue buttons go in one circle, red buttons in a second circle, and the buttons that have both red and blue will go in the overlapping section between the two circles. Or, you can begin crossing different attributes- you could sort buttons into green, 3 holes, and round. Different buttons will fit into different sections of the diagram- where will the green star with 2 holes go? what about the round, blue button with 3 holes? the round, green one with 3 holes?

*The next level of categorization is to identify sorted groups and then think about what the groups will look like if you subcategorize them or mix them in certain ways. This is where students begin thinking about subsets (a smaller group within a category), unions (combining of two categories), and intersections (what two different categories have in common). Really, this is an articulation of what they are drawing out on the Venn diagram, but as the concepts move from categories of items to categories of geometric attributes or categories of number types, the complexities increase. A Venn diagram showing categories and intersections of polygons, prisms, angle measures, etc. could have value into high school. Sorting numbers into sets, subsets, unions, and intersections- integers, prime numbers, factors of 26, multiples of 75- has applications in number theory into high school and beyond, depending on your field of study.

Toldot- Base-10 and Number Sense related activity ideas

"Yitzchak sowed in that land, and in that year he reaped a hundredfold; thus had Hashem Blessed him." ~Bereishit 26;12

Rashi on 26;12-
"A Hundredfold- For they assessed [the field] to determine how much it is fit to produce and it produced for every unit they estimated it could produce a hundred units. Our Rabbis said, This assessment was done for tithes."

In this section of the parsha, we learn of Yitzchak's time living in Gerar, under the rule of Avimelech. Bereishit Rabbah (64;6) asks: if we're not supposed to find a blessing in a measurable quantity, how could this crop be considered a blessing? You are not supposed to count on a crop before it is harvested (like the old adage- "Don't count your chickens before they hatch"). How does this make sense? The Mizrachi explains that the crop was traditionally estimated in order to make an advanced calculation for tithings to be easily set aside. The midrashim (Tosafot HaShalem citing Rivah) teach us that this crop that is referred to was sowed during a time of famine in the land. Yitzchak had estimated what his tithe should be in order to be able to set it aside for the poor as soon as possible, and he had estimated much less of a crop due to the famine in the land.  It is with this understanding that we see how this was such a blessing for Yitzchak. 

Activity Connections:
There are two primary mathematical concepts that connect to this crop:
1) the concept of calculating and setting aside tithings
2) the concept of one hundredfold 

Interestingly enough, both of these concepts connect to powers of 10. 
*Tithings are classically 1/10 of a crop (the root of the hebrew word- מעשרות/מעשר- is actually related to עשר, the number 10)
*one hundredfold means times 100, which is a power of 10 (10²;or X10 and then X10 again)

Activity ideas in increasing conceptual difficulty:
*What does a tithing or 1/10 of a group really look like? For the youngest students, they could begin by just taking a group of items and separating the items into 10 equal groups. Having a set-up of 10 cups, bowls, or plates to sort the pieces into will help students keep track of their sorting. This is doable even for preschool children. After they divided their items into 10 piles, they can count each pile to make sure that they all have the same number of items in them. From there, they can act out separating 1 of the piles to give away and keeping 9 of the piles for themselves. For beginners, you want to simplify the process for them by making sure that the number of items they start with is a multiple of 10 (so that it divides equally into 10 piles).

*As a step up from the previous activity, you can have students divide larger groups of items, and incorporate estimation or calculation of what amount the tithing will actually be when it's separated out. Students who have had exposure to the concept of fractions and decimals could also estimate and calculate the value of tithings for amounts that are not strictly multiples of 10.

*One hundredfold- what does x100 really look like? What if Yitzchak had anticipated having 5 bundles of grain? How much did he actually have that year? What if he had anticipated 10 bundles of grain? Younger students can physically count or draw out the difference of a group of 5 and a group of 500. Small or medium sized graph paper can be good for drawing out a model of these differences (with each box modeling 1 unit). If students have the dexterity, small manipulatives like paper clips can also be good. For younger students, these can be modeled with blocks or Legos, but you need to have a good supply in order to model the hundreds. (Tip: Creating stacks or grouping of 10 to count everything out will help make counting easier and will help children develop their number sense.)

*Older students can graph the difference between the anticipated crops and the actual crops. Is there a pattern to the difference? How would the difference be expressed as an algebraic expression?

*Another activity for older students would be a comparison of the tithings between the anticipated crops and the actual crops. What would a graph of these differences look like? How could the differences be expressed algebraically? How does the pattern of the difference in tithings (anticipated vs. actual) compare to the pattern of the difference in crops (anticipated vs. actual)? How do the algebraic expressions of each comparison compare to each other?

Chayei Sarah- Currency related activity ideas

A note to readers- Having written a full cycle of parsha posts, I've decided to try a slightly different approach. For this next cycle, I'm beginning with identifying a concept and then posing concept specific questions and activities for investigating the concepts at varying academic levels- still K-8 level.  I'd love feedback from readers as I keep the process going. Are you just an interested reader? Am I still keeping you interested? Are you a teacher/parent looking for ideas? Which ideas do you find most helpful? Have you tested out any ideas with your students/children? Looking forward to your feedback!
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"...And Ephron replied to Abraham, saying to him: 'My lord, hear me! Land worth four hundred silver shekalim; between me and you- what is it? And bury your dead.' Avraham listened to Ephron, and Avraham weighed out to Ephron the money that he had mentioned in the hearing of the children of Heth, four hundred silver shekalim in negotiable currency. And Ephron's field, that was in Machpelah, that was facing Mamre stood- the field and the cave within it and all the trees in the field, within its entire boundary all around- as Avraham's as a purchase in the view of the children of Heth, with all who came to the gate of his city. And after that, Avraham buried Sarah his wife in the cave of the field of Machpelah facing Mamre, which is Hevron, in the land of Canaan." ~Bereishit 23;14-19

Activity Connections:
The two ideas that stand out to me from this passage are:
1) Avraham's weighing out of the money
2) The fact that the Torah specifically says that he was using "negotiable currency"

For some previous thoughts about scales and balance, see my post on Parshat Ki Teitzei where I talk about weights and measures.

Rashi on 23;16 explains that "negotiable currency" is stated because the shekel that the Torah usually refers to is a smaller shekel (equal to 1 sela). In our case, though, Avraham chose to use shekels which are acceptable as currency even in areas that use larger shekels ("kanterin", which are equal to 100 selas). 

Activity ideas in increasing conceptual difficulty:
*Discussion and exposure to local current-day currency; discussion around equivalent currencies
eg. for US currency, exposure to pennies, nickels, dimes, quarters, half-dollars, dollar coins, dollar bills, etc and discussion/practice with the value of each and equivalencies (5 pennies = 1 nickel; 25 pennies = 1 quarter = 5 nickels = 2 dimes + 1 nickel)

*Combine currency with the idea that Avraham weighed out his money. Does this still work with current day currency? Will 5 pennies balance with 1 nickel on a balance scale? Our current money is more of a representation of value than true equivalence between coin sizes, weights, and values.

*Look at the difference between what 400 shekalim would have been if they were each 1 sela instead of being 1 kanterin. This can be done at varying levels for different ages and abilities. What does it physically look like using manipulatives to represent 1 sela? How could you calculate how many selas Avraham actually paid Ephron? 

*If you look at the comments section in last year's post on Chayei Sarah, you'll find a link to some calculations on current day equivalence for a shekel (one that's the value of a sela). How can you use this information to calculate the current day value for what Avraham actually paid for Maarat HaMachpelah? How different is that from what he would have paid in current day currency if he used selas instead of kanterin?

*How does the amount of money paid toward Maarat HaMachpelah compare to the value of Rivkah's dowry (the bracelets and nosering discussed in last year's post)? How does a beka compare to a sela? to kanterin?

*If you compare the difference between values of selas and kanterin, is the difference linear growth or exponential growth? (Hint: 1 kanterin = 100 selas; 2 kanterin = 200 selas; and so on)

Vayeira- Non-standard Units of Measure

"The water of the skin was finished, and she cast off the boy beneath one of the trees. She went and sat herself at a distance, some bowshots [away], for she said, 'Let me not see the death of the child.' And she sat at a distance, lifted her voice, and wept." ~Bereishit 21;15-16

Non-Standard Units of Measure:
We are all familiar with standard units that we use to measure on a regular basis- we measure length with inches, feet, miles, centimeters, meters, kilometers; we measure liquids with cups, pints, quarts, gallons; we measure weight with ounces, pounds, grams, kilograms. Sometimes, though, the standard units that we use either aren't available or aren't necessary for the purposes of what you're doing.

When children are first learning about measuring, a common activity is to choose a standard by which to measure different items, and then students can compare the items based on their measurements. For example, they may use stacking cubes (all of the same size) to measure items in their surroundings- a pencil, a chair, a water bottle, etc. They can record their findings and then compare which items are shorter and which items are longer (or taller). As long as the same standard is used throughout an activity, all the items measured can be compared to each other.

Anything that you choose can be used as a non-standard unit for measuring- a paper clip, a pencil, a shoe- as long as the repeated measurement is using an item (or items) that are the same exact length. So, you can't use a mixture of large and small paper clips to measure items, but you can choose one or the other. You can see how many water bottles it takes to fill up a pitcher, but you need to fill the water bottles full each time to establish a temporary standard for your measurements. 

If you want, or need, you can always convert your non-standard measurements into a standard unit afterwards by measuring your non-standard unit. If your paperclip is 3/4 of an inch long, you have enough information to convert your measurements into inches. If your water bottle holds 2/3 C of water, then you can calculate how many cups your pitcher holds.

Parsha Connection:
In this week's parsha, we learn of Hagar and Ishmael's wandering after they are sent away from Avraham and Sarah's home. We are told that when Hagar believes that Ishmael is dying, she sits "some bowshots" from him, so as not to have to watch him die. What does this mean practically, for our understanding of the distance between Hagar and Ishmael at this time?

Bereishit Rabbah 53;13 explains two aspects of this in order for us to understand the distance:
1) first it explains that since it is written "bowshots", plural, but with no specific quantity, it must mean that the minimum plural is implied. Since the smallest plural number is 2, therefore it's 2 bowshots away.
2) second it explains that a standard bowshot during this time was 2,000 cubits, or a little more than today's measure of 1/2 a mile.

Based on this explanation, we can use the parsha's non-standard measurement of a bowshot to calculate that, by current day measurements, Hagar was approximately a distance of a mile away (or just over a mile) from Ishmael during this time.

Everyday Connection:
Let's say you have a recipe for a sauce that calls for equal parts of each ingredient. You don't need to measure exactly with a 1 cup measure- if you have a cup without measurement markings, you can still use that cup to measure out your ingredients. All you need to do is make sure that you measure each ingredient to the same point on the cup, and you'll have equal parts of each ingredient.

What if you need to measure the length of an item, but you don't have a ruler nearby? You can use a pencil, a piece of paper, or even your shoe- count how many "shoes" long your item is, and then you can measure the length of your shoe later to convert your measurement to a standard unit.